Wilcoxon Signed-Rank Test · Critical Values

Parameters

n (number of non-zero differences)

Larger n approaches the normal approximation; this tool still computes the exact distribution (default max 60; you can raise it in the code). Test (tail)

Table range (n)

Default table: n = 6–20 (matches the original script).

Statistic

W = min(W⁺, W⁻), where W⁺ is the sum of positive ranks and N = n(n+1)/2.

Performance notes

The original Python version enumerated all 2ⁿ sign configurations. This implementation uses dynamic programming to compute the exact distribution of W⁺ (subset-sum counts), then maps it to W = min(W⁺, N−W⁺):

Time: from O(n·2ⁿ) down to O(n·N), where N = n(n+1)/2.
Counts use BigInt to avoid overflow; probabilities are compared as exact fractions.
Cached per n: the distribution is computed once and reused.

Results

Total rank sum N

—

Critical value W_crit

—

P(W ≤ W_crit)

—

Distribution (default: PMF P(W=w))

Show CDF

Note: Column headers show “one-tailed α / two-tailed α”. Since the statistic is W=min(W⁺,W⁻), tables are often published for two-tailed α; for a one-tailed test, you can use the corresponding two-tailed 2α threshold.